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ALGEBRA OF COMPLEX NUMBERS

Introduction

A number of the form z=a + ib, i=√-1, where ‘a’ and ‘b’ are real numbers is called a complex number. Here ‘a’ is the real part and ‘b’ is the imaginary part of ‘z’.

The real part is denoted by Re z and imaginary part by Im z.

If the real part is zero then is is said to be purely imaginary and if the imaginary part is zero then it is said to be purely real.

All real numbers are complex numbers with imaginary part zero.

For example: 5= 5 + i0 -3 =-3 + i0Two complex numbers z1=a + ib and z2=c + id are equal if a=c and b=dIf two complex numbers are equal then their real parts are equal and imaginary parts are also equal.Example: If 4x+ i(4y-x) = 8+ i6, then find ‘x’ and ‘y’Solution: Since the complex numbers are equal, we have 4x=8 and 4y-x=6 4x=8 x=2 4y-x =6 becomes 4y-2 =6

4y=6+2=8 y=2Hence x=2 and y=2

Algebra of Complex Numbers

In algebra of complex numbers we deal with addition of two complex numbers, subtraction of two complex numbers, multiplication of two complex numbers , division of two complex numbers and the properties satisfied by them.

Addition of two complex numbers

Let z1= a + ib and z2= c + id be any two complex numbers, then the sum z1+z2 = (a+c) + i(b+d), which is again a complex number. While adding two complex numbers, we have to add real parts together ad imaginary parts together.Properties satisfied by addition of complex numbers.

Closure Property: The sum of two complex numbers is a complex number.

Commutative Property: For any two complex numbers z1 and z2, z1+z2= z2+z1

Existence of multiplicative identity: To every complex number z, there exists 1_i0 (denoted as 1), called the multiplicative identity such that z.1 = 1.z =z.

Existence of multiplicative inverse: For every non-zero complex number z we have 1/z or z-1, called the multiplicative inverse of z such that z.1/z=1

Distributive Law: For any three complex numbers z1, z2, z3

a) z1(z2+z3) = z1z2 + z1z3b) (z1+z2)z3 = z1z3 + z2z3

Division of two complex numbers

Given any two complex numbers z1 and z2, where z2≠0, the quotient z1/z2 is defined by z1. 1/z2Here we multiply by the conjugate of the denominator to get the answer.Example: Find the multiplicative inverse of √5+3iSolution: The multiplicative inverse of √5 + 3i is Hence the multiplicative inverse of √5 + 3i is (√5/14) + i(3/14)

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